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 Post subject: Nested integrals and limitPosted: Thu Nov 12, 2015 3:15 pm

Joined: Sat Nov 07, 2015 6:12 pm
Posts: 841
Location: Larisa
I am posting this exercise here but perhaps it can fit elsewhere.

Show that the integral:

$$\begin{eqnarray*}V_n=\int_0^1 \int_0^1 \cdots \int_0^1 \frac{x_1^2+x_2^2 +\cdots +x_n^2}{x_1+x_2+\cdots+x_n}\, {\rm d}x_1 \, {\rm d} x_2 \cdots \,{\rm d}x_n \end{eqnarray*}$$

converges to $2/3$ as $n \rightarrow +\infty$ and that the product $n(V_n -2/3)$ remains bounded.

Source

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